Keywords: Automotive, Stability, Numerical simulation
Abstract: Remote driving has become a prevailing research topic due to its advantages, such as reducing the cost of on-board autonomous systems, utilizing more computation power, and providing an alternative solution to autonomous driving under critical circumstances. However, relying on the communication networks to transmit information and control commands introduces time delays into the system, in addition to the existing actuation delay on the vehicle side and the processing delay of the remote operator.

In this work, we present the effects of time delays on remote driving for the case of following a curved path. We demonstrate that the longitudinal velocity, the overall delay and the path curvature all affect the performance of remote maneuvers. We introduce the dynamical model of a teleoperated vehicle following a curved path. We then study the linear stability of the lateral dynamics while the teleoperated vehicle follows a path of constant curvature, accounting for the speed and the overall teleoperated network delay. We present simulation results for different maneuvers in a parking lot scenario and relate the numerical simulations to the analytical stability boundaries.

Keywords: Automotive, Control of linear systems, Stability
Abstract: In this paper, we demonstrate the detrimental effects of latency in remote driving with an example of straight-path following. To address the instability and performance degradation caused by the latency in the remote driving control loop, we propose to use an act-and-wait strategy on top of the existing controller. This strategy can potentially stabilize the system under large latency using the original control gains, and achieve dead-beat control with modified control gains. Stability and performance analysis is conducted for the act-and-wait strategy to provide insights on modulating the control input under large latency.

Keywords: Control of nonlinear systems, Automotive, Infinite dimensional systems
Abstract: The reduced order dynamics of a car-following model is investigated when an automated lead vehicle provides guidance to a human driver. Considering the relevant time delay, the system dynamics is governed by a delay differential equation, which has an infinite dimensional spectrum. The concept of spectral submanifolds is extended to time delay systems, with the help of which model-order reduction is carried out to project the dynamics from the infinite dimensional state space to a low dimensional invariant manifold and obtain the relevant system dynamics. The spectral submanifold calculation is carried out both for a real dominant eigenvalue and for a pair of complex conjugate dominant eigenvalues of the linearized system.

Keywords: Automotive, Nonlinear dynamics, Stability
Abstract: The loss of vehicle stability is a major cause of road accidents, with over one million fatalities recorded yearly in the world. A complete mathematical characterisation of the behaviour of road vehicle-and-driver under the action of severe perturbations, e.g. wind gusts or evasive maneuvers, is still lacking. Studies on global stability of road vehicle-and-driver are in their infancy, offering numerous avenues for enhancing the active safety of vehicles, whether human-driven or automated. This paper aims to start providing a substantial contribution in the field. Considering a simple vehicle-and driver model, a subcritical Hopf bifurcation is found at high forward speed, caused by the driver’s delayed steering action. Both rectilinear and circular motions of the system are characterized by a saddle-type limit cycle. A region of attraction of stable equilibria of the system is thus computed, both numerically and with experimental tests performed at the driving simulator.

Keywords: Nonlinear dynamics, Stability, Control of nonlinear systems
Abstract: The dynamics of electric scooters is analyzed via a spatial mechanical model. The e-scooter is balanced at zero speed, by applying internal driving torque to the front wheel. A hierarchical, linear state feedback controller is designed with feedback delay. The linear stability charts of the delayed controller are constructed with semi-discretization. The effects of the feedback delay and the center of gravity position of the handlebar on the linear stability are investigated. The performance of the control algorithm is tested by numerical simulations with the nonlinear governing equations.

Keywords: Stability, Automotive
Abstract: In this study, the reverse motion of an autonomous truck–semitrailer combination along a circular path is investigated. To do this, the single-track kinematic model is applied. As the time delay in the control loop (assuming a simple proportional control law) is considered, the semi-discretization method is used for the stability analysis. The significant effect of the time delay on stability is emphasized with nonlinear simulations.

Keywords: Control of nonlinear systems, Infinite dimensional systems, Biology
Abstract: In this paper, the sampled-data stabilization problem under safety constraints is studied for nonlinear time-delay systems. In particular, a methodology for the design of sampled-data safe stabilizers is provided for nonlinear systems affected by state delays. As a first step, the new notion of Safe Steepest Descent Feedback (SSDF), continuous or not, is introduced by suitably combining the well-known notions of Steepest Descent Feedbacks and Barrier functions considered to cope with safety requirements. Then, the stabilization in the sample-and-hold sense theory is used as a tool to show the existence of a suitably fast sampling such that: the digital implementation of SSDFs, continuous or not, ensures the semi--global practical safe stability property of the related closed-loop system with arbitrarily small final target ball of the origin. In the theory here developed, time-varying sampling periods are allowed. The proposed theoretical results are validated through an application concerning the plasma glucose regulation problem in Type-2 diabetic patients via artificial pancreas.

Keywords: Control of linear systems, Infinite dimensional systems, Robustness
Abstract: This paper proposes a way to efficiently implement stabilizing controllers for delay systems of the retarded type. We indeed choose a particular controller (central controller) in the set of all stabilizing controllers and decompose its transfer function in terms of stable delay systems and Finite Impulse Response (FIR) filters. The central controller is written in such a way that the corresponding Matlab/Simulink implementation scheme is obvious. We illustrate that all stabilizing controllers can be implemented by using stable components only around the central controller. We also discuss stability robustness in the presence of implementation errors. The method is illustrated in an example.

Keywords: Control of linear systems, Stability, Infinite dimensional systems
Abstract: In this paper, we address the observer based control problem for SISO (Single-Input/Single-Output) LTI (Linear Time Invariant) system subject to delay in the measurement data. The proposed design approach includes two steps: first, by using a partial pole placement approach based on the MID (Multiplicity-Induced-Dominancy) property, we construct a delay-based controller guaranteeing the stability of the closed-loop system. Second, we exploit the control gain to determine the observer-controller matrices. The effectiveness of the proposed methodology is illustrated through the problem of stabilizing a double integrator and of a wind-tunnel system with delayed feedback.

Keywords: Control of linear systems, Nonlinear dynamics, Stability
Abstract: This work introduces a new class of reset linear systems, called reset-delayed linear systems, which consists of multivariable dynamical systems featuring a continuous-time state evolution interrupted by discontinuities of some, or even all the state variables at isolated time instants. In particular, these discontinuities abide by an algebraic equation imposing that the state variables involved take the same values they respectively had a certain amount of time before the time instant when the discontinuity is triggered. In this way, the evolutions of the state variables involved turn out to be delayed by the amount of time considered in the reset operation. In the presence of suitably chosen forcing actions, reset-delayed linear systems can effectively model repetitive behaviors which imply a discontinuity at the junction between one cycle and the subsequent one. Herein, some structural properties of this class of multivariable linear dynamical systems are studied and the geometric notions of invariance and controlled invariance are formalized and applied to the solution of the disturbance decoupling problem.

Keywords: Control of linear systems, Robust control, Stability
Abstract: In this study, we consider the stable H-inf controller design problem for multi-input multi-output retarded-type time-delay systems with multiple point-wise time delays. For the controller synthesis, an eigenvalue optimization-based design method is proposed in which stabilization and robust control problems are formulated as the minimization of spectral abscissa and H-inf norm of the associated closed-loop system, respectively, under the constraint of controller stability.

Keywords: Control of linear systems, Consensus problems and synchronization, Stability
Abstract: We present control design tools where a class of delay-based controllers are subject to unintentional delays, in the context of linear time-invariant systems.

Keywords: Spectrum computation and optimization, Control of linear systems
Abstract: TDS-CONTROL is an integrated Matlab package for analysis and controller design of linear time-invariant systems with discrete delays, of retarded and neutral type. These systems are assumed to be given in state-space form, although functionality is provided to obtain such a representation from a frequency-domain description. Firstly, the package offers various methods for analysis. More specifically, it contains methods for computing the spectral abscissa, the H-infinity norm, the pseudospectral abscissa, and the distance to instability. As TDS-CONTROL is designed with neutral equations in mind, it has the appealing feature that the sensitivity of the spectral abscissa and H-infinity norms with respect to small delay perturbations is taken into account. Secondly, TDS-CONTROL contains functionality to design fixed-order output feedback controllers. The respective design algorithms are based on optimizing the spectral abscissa, the H-infinity norm or a combination of both with respect to the elements of the controller matrices. For example, to design a (strongly) stabilizing controller the (strong) spectral abscissa is minimized. Furthermore, this optimization-based approach allows to easily impose structure on the controller, which enables the design of decentralized, overlapping and PID controllers. Finally, by allowing the plant to be described in delay descriptor form, Pyragas-type, acceleration and delay-based feedback controllers can be considered.

Keywords: Control of linear systems, Stability, Infinite dimensional systems
Abstract: In recent years, methodological progress has been made in the partial pole placement for infinite-dimensional systems. In particular, new control methodologies have been drawn up exploiting past data to improve the stability of such dynamical systems. A dedicated Python software called Partial Poles Placement via Delay Action (P3delta) has been developed. This brief note lists the new development directions that are planned and will be integrated into P3delta within the framework of internships for graduate students.

Keywords: Modeling and identification, Nonlinear dynamics, Approximation techniques and numerical methods
Abstract: The infinite-dimensional feature of the time delay systems results in complex behavior and it is often difficult to build models for those systems using first principles. Data-driven methods have shown their power in modeling dynamical systems and advanced training algorithms have been developed in the past decades. In this abstract, we aim to bridge between time delay neural networks and delayed dynamical systems from continuous-time perspective. We demonstrate that with explicit delay in the input layer and data containing rich dynamics, the trained network has better interpretability and generalizability. We present a training method and demonstrate that the delays and nonlinearities can be learned simultaneously with single high-quality trajectory. We propose to use the maximal Lyapunov exponent (MLE) to represent the richness of the training data.

Keywords: Nonlinear dynamics, Control of nonlinear systems, Numerical simulation
Abstract: We construct a deep neural network from a single nonlinear node with delayed feedback loops and time-varying feedback strength. For piece-wise constant control signals, this system implements a multi-layer perceptron. This single neuron setup significantly reduces hardware requirements and enables the implementation of large neural networks through temporal dynamics. We call this approach Folded-in-time DNN (Fit-DNN). We present a modified back-propagation algorithm for training Fit-DNNs, demonstrating its effectiveness across benchmark tasks. Additionally, we show how one can train time-continuous signals by applying ideas from optimal control theory of systems with multiple delays.

Keywords: Modeling and identification, Filtering and estimation, Approximation techniques and numerical methods
Abstract: For a reaction-diffusion equation with unknown right-hand side and non-local measurements subject to unknown constant measurement delay, we consider the nonlinear inverse problem of estimating the associated leading eigenvalues and measurement delay from a finite number of noisy measurements. We propose a reconstruction criterion and, for small enough noise intensity, prove existence and uniqueness of the desired approximation and derive closed-form expressions for the first-order condition numbers, as well as bounds for their asymptotic behavior in a regime when the number of measurements tends to infinity and the inter-sampling interval length is fixed. We perform numerical simulations indicating that the exponential fitting algorithm ESPRIT is first-order optimal, namely, its first-order condition numbers have the same asymptotic behavior as the analytic ones in this regime.

Keywords: Modeling and identification, Nonlinear dynamics, Approximation techniques and numerical methods
Abstract: We present a pragmatic approach to the sparse identification of nonlinear dynamics for systems with discrete delays. It relies on approximating the underlying delay model with a system of ordinary differential equations via pseudospectral collocation. To minimize the reconstruction error, the new strategy avoids optimizing all possible multiple unknown delays, identifying only the maximum one. The computational burden is thus greatly reduced, improving the performance of recent implementations which work directly on the delay system.

Keywords: Modeling and identification, Nonlinear dynamics
Abstract: We investigate the capabilities of the Sparse Identification of Nonlinear Dynamics (SINDy) method for time delay systems. We generate a dataset based on a numerical simulation and present two methods to identify the system with time delays. First, we show the threshold iteration using the STLSQ algorithm. Second, we show a delay iteration method to reduce the dimension of the candidate terms.

Keywords: Consensus problems and synchronization, Networked and multi-agent systems, Decentralized and distributed control
Abstract: This article addresses the distributed average consensus problem for cooperative-antagonistic Multi-Agent Systems (MASs) in the presence of multiple and unknown communication time-varying input delays, whose actual values depend on the effective conditions of the wireless channels. A distributed delayed control strategy, able to counteract the unavoidable communication impairments, is proposed for the achievement of the exponential stability of the networked control system. The convergence analysis, which leverages the Lyapunov stability and Halanay's lemma, also handles the fast time-varying delay case and analytically revels the relationship among the estimation of the maximum delay upper-bound, the smallest nonzero eigenvalue of Laplacian matrix, i.e. the Fiedler eigenvalue, and the control gain. Finally, numerical simulations confirm the theoretical derivation.

Keywords: Consensus problems and synchronization, Networked and multi-agent systems, Decentralized and distributed control
Abstract: This paper delves into the challenge of achieving group consensus in a multi-agent network with both input and communication delays. In the absence of any restrictive graph structure, the concepts of primary and secondary layer subgraphs are utilized to identify the number of naturally emerging consensus groups and their members within the network. Furthermore, delay conditions are derived to guarantee the system’s convergence to group consensus. Numerical examples are provided to demonstrate the validity of the theoretical results.

Keywords: Networked and multi-agent systems, Decentralized and distributed control, Physics
Abstract: While distributed control solutions enhance modularity, scalability, and robustness of Microgrids (MGs), compared to traditional centralized control approaches, their synthesis and analysis are more complicated due to the insertion of shared communication networks in feedback control loops, which enable the information exchange among spatially Distributed Generations (DGs). Thus, several network-induced communication constraints may arise during data transmission/acquisition, which can compromise the global stability of the entire MG. Therefore, this work introduces a unified resilient distributed control approach to solve the voltage regulation problem in inverter-based islanded MG undergoing both uncertain imperfection-prone communication links and heterogeneous time-varying delays whose co-presence, to the best of authors knowledge, has not been considered before. It is worth mentioning that the modeling of link uncertainties exploited through this work can cover different classes of communication failures and data manipulation attacks. The voltage exponential stability of the entire MG is analytically proven by exploiting the Lyapunov-Krasovskii theory, which provides stability conditions in terms of feasible Linear Matrix Inequalities (LMIs) depending on both maximum admissible delays and communication links uncertainties upper-bounds preserving the stability. Numerical results confirm the effectiveness and the resilience of the theoretical derivation.

Keywords: Traffic flow analysis and control, Nonlinear dynamics, Networked and multi-agent systems
Abstract: This work studies the nonlinear dynamics of vehicular traffic on highways, including human-driven vehicles (HVs) and connected automated vehicles (CAVs). Highway traffic is susceptible to the onset of traffic jams that significantly reduce the efficiency of transportation. With automation and connectivity technologies, CAVs hold the promise of providing aid to this problem. By demonstrating smoother driving behavior than HVs, CAVs could make the overall traffic smoother and mitigate traffic congestion. To accomplish this goal, however, the effect of CAVs on the traffic dynamics must be understood and their controllers must be designed so that they positively impact the traffic flow. In this work, we conduct stability and bifurcation analysis for the nonlinear delayed dynamics of mixed traffic systems. We design controllers for CAVs that make the equilibrium of the underlying dynamics globally stable. With sufficient penetration of CAVs in the traffic flow, this ultimately helps to avoid traffic jams, leading to more efficient transportation on highways.

Keywords: Prediction based control, Automotive
Abstract: In this paper we present a prediction-based Cooperative Adaptive Cruise Controller for vehicles with actuation delay, applicable within heterogeneous platoons. We provide a stability analysis for the discrete-time implementation of this controller, which shows the effect of the used sampling times and can be used for selecting appropriate controller gains. The theoretical results are validated by means of experiments using full scale vehicles.

Keywords: PID control, Control of linear systems
Abstract: The design of the PI controller for stable linear plants based on the magnitude-optimum criterion can be applied directly for high-order linear models with dead time, without need of the model reduction. This paper discusses key properties of this method, especially considering oscillating plants. Even though the closed-loop response is specified only for low frequencies and the robustness requirements are not explicitly formulated in the design objective, in the case of plants without zeros a large closed-loop stability margin is ensured automatically regardless the dead time value, under the assumption that the damping coefficients corresponding to the oscillatory modes of the plant are sufficiently high.

Keywords: PID control, Spectrum computation and optimization, Infinite dimensional systems
Abstract: Dimensional analysis approach to describing a higher-order system with input delay is applied to formulate the dominant three-pole placement for tuning the PID controller. First, the PID control admissibility analysis is provided for a system higher order in the dimensionless parameter space. Achieving the admissibility conditions the admissible dominant pole placement is performed when a trio of poles is prescribed to change the higher order dynamics. In case of the third order system dynamics, the rightmost zeros of the fourth-degree characteristic quasi-polynomial are identified with the prescribed trio of poles while in case of the fourth-order system dynamics, the rightmost zeros of the fifth-degree characteristic quasi-polynomial are identified with the prescribed trio of poles. Finally, the PID controller gains are mapped in the dimensionless three-pole parameter space showing proper selection of the trio of poles to be assigned for fast and well-damped control responses.

Keywords: Vibration control, Numerical simulation, Stability
Abstract: This contribution presents novel analytical approximate closed form expressions used together with the semidiscretization method to approximate the finite-time stability boundaries of delay differential equations with time varying parameters. As a proof-of-concept, the formulae are tested using the simple example of the single degree of freedom undamped delayed oscillator and compared extensively to Matlab’s dde23 numerical scheme. The presented framework can be used generally in delayed systems.

Keywords: Regenerative chatter, Stability, Lyapunov functionals and Lyapunov matrices
Abstract: This work aims at determining stability regions for turning processes in the plane of the parameters: spindle speed and depth of cut, based on both time and frequency domain techniques. For the time domain approach, a Lyapunov-Krasovskii functional is employed, while for the frequency domain approach, the D-partitions method is applied. An interesting result of this work is that the critical depth of cut can be identified from both approaches and it is consistent with the Tlusty's formula derived from the Nyquist criterion.

Keywords: Neutral and advanced equations, Stability, Lyapunov functionals and Lyapunov matrices
Abstract: Necessary and sufficient stability conditions for neutral-type linear time-delay systems are presented. Our approach is based on discretizing functionals with prescribed derivative for this class of systems via the discretized Lyapunov functional method introduced in Gu (1997). As a result, the discretized functional is expressed by a quadratic form, whose inner matrix is expressed in terms of the delay Lyapunov matrix. Remarkably, the non-negativeness of this matrix is connected with the positive definiteness of the matrix presented recently in Gomez et al. (2019). Along with the estimation of the functional approximation error on a special set of functions, this provides a stability criterion.

Keywords: Infinite dimensional systems, Stability, Vibration control
Abstract: It is considered a mechanical system composed of a string in cantilever having a pointwise damping. The discontinuous B(oundary) V(alue) P(roblem) is split in two coupled BVPs. Introducing the Riemann invariants and taking into account their invariance along the characteristics, a system of continuous time difference equations with two delays is associated to the coupled BVPs and a one-to-one correspondence between the solutions of the two mathematical objects is proven. Then a theorem on exponential stability for the difference system is obtained. The stability depends on the arithmetic properties of the delay ratio. The aforementioned one-to-one correspondence is then used to establish exponential stability of the classical solutions of the coupled BVPs. Some conclusions and a sketch of open and future research subjects end the paper.

Keywords: Robust control, Control of linear systems
Abstract: Bounded real lemma is discussed for commensurate delay systems, and a necessary and sufficient condition for the achievable H-infinity performance with respect to the interval delay is characterized by parameter-dependent LMIs. The condition is further transformed to standard LMIs, and it is shown that the parameter-region partitioning enables to treat the necessary and sufficient condition with arbitrary resolution and precision. The computation efficiency for the evaluation of optimal H-infinity performance is discussed with numerical examples.

Keywords: Vibration control, Infinite dimensional systems, Numerical simulation
Abstract: This paper focuses on the transverse vibration midpoint control of a string. More precisely, we explore the use of a control block that includes delays in both input and output signals under the assumption that the memory-window length is also a parameter of the control law. Relatively simple to construct, such a control law was successfully used in the boundary control of the transport and wave equations. The effectiveness of the proposed method is illustrated through numerical simulations and the corresponding finite-difference scheme is detailed.

Keywords: Infinite dimensional systems, Vibration control, Neutral and advanced equations
Abstract: Repetitive control represents a control scheme aimed at following or rejecting periodic exogenous signals. Its essence relies on incorporating a time-delayed signal model capable of expressing every periodic signal with a known period. Nevertheless, stabilization becomes a major issue due to the dynamics introduced to the loop by the signal model leading to delay-differential equations of neutral type for the closed-loop system. This paper briefly reviews the elements of the original repetitive control concept and the underlying issues from a time-delay system theory viewpoint. Later on, based on the highlighted facts, potential modifications to make the control scheme practically applicable are explored and briefly presented.

Keywords: Modeling and identification, Control of linear systems, Physics
Abstract: In the talk we consider a class of weakly time-delayed planar linear differential systems with constant coefficients. It is shown that general solutions of such systems can be written in a closed form, depending on the eigenvalues of the matrix of linear non-delayed terms and have a simple behavior when a transient interval is passed. Although such systems cover only a narrow set of linear differential systems with delay, they are of certain interest. This is documented by a wind-tunnel model with time-delay.

Keywords: Adaptive control, Compensation schemes, Control of linear systems
Abstract: The paper considers the problem of adaptive compensation of external disturbances affecting LTI SISO plant with the input delay and unmeasurable state. The plant to be controlled can be unstable. The solution is based on a suitable disturbance parameterization and prediction as well as on the use of special swapping scheme allowing one to overcome the problem of the input delay in adaptation algorithm. A new control law with the property of almost finite-time adaptive compensation is proposed. The main results are rigorously proved and illustrated by simulation.

Keywords: Regenerative chatter, Stability, Vibration control
Abstract: An industrial physical modeling of the boring bar machine system is compiled in this work. The machine dynamics is originated from field measurements by having corresponding frequency response functions (FRFs), which are known to be difficult to model reliably. The dynamics model is achieved by using receptance coupling substructure analysis (RCSA). The corresponding state dependent delay differential equation (SD-DDE) model of the boring operation is developed.

Keywords: Modeling and identification, Biology, Approximation techniques and numerical methods
Abstract: Many successful control strategies for anesthetic processes do not adequately consider time delays despite potential 30-second delays from biophase and bispectral index (BIS) monitoring. This study introduces linear delayed Pharmacokinetic-Pharmacodynamic (PK-PD) patient models that correlate more to real patient outputs than common non-delayed models. The delayed models are obtained for both effect-site concentration (C_e) and BIS outputs. The model structures are identified using autoregressive exogenous input (ARX) and delayed-ARX (DARX) polynomial models and least squares estimations (LSE) using the real surgical data of the open-access database VitalDB. Validation of delayed PK-PD patient models involves using correlation analysis. Results indicate that PK-PD patient models with output delays exhibit higher correlations with real patient data, particularly when considering BIS output.

Keywords: Approximation techniques and numerical methods, Nonlinear dynamics, Neutral and advanced equations
Abstract: A delay differential algebraic system of equations describing the dynamics of a pressure relief valve was obtained. The time delay mathematical model, where the delay term is related to the wave propagating in the outlet pipe, is convenient for stability analysis; however, the nonlinear analysis is challenging. The algebraic equation contains a square root nonlinearity, thus, the system is implicit and, moreover, neutral. In previous studies, limit cycle branches were calculated via the continuation method with singular perturbation, but the reliability of the approximate solutions must be proven. In this study, direct numerical simulations support the previously depicted bifurcation scenario, specifically, the presence of the fold bifurcation based on the transient dynamics.

Keywords: Nonlinear dynamics, Robustness, Approximation techniques and numerical methods
Abstract: The dynamical integrity, or robustness against external perturbations, is a practically important aspect of nonlinear dynamical systems typically hard to address. This research introduces an algorithm to assess the dynamical integrity of steady states in time delay systems. More precisely, the method estimates the local integrity measure, which is the radius of the largest hypersphere centered in the solution and entirely included within its basin of attraction. The algorithm does not account for potential fractal boundaries and it offers a practically relevant and computationally efficient measure.

Keywords: Approximation techniques and numerical methods, Nonlinear dynamics, Stability
Abstract: The subject is a continuous-time second-order damped oscillator whose restoring force is nonlinear and delayed. It is otherwise called the "sunflower equation." The goal is to find a forward invariant set containing the equilibrium, as well as to estimate its domain of attraction. The aim is for delay-dependent results. We formulate and apply the method of delay-free comparison systems. Using a reasoning similar to the classical Bendixson pocket principle, we develop an invariant set estimation algorithm based on the numerical integration of the comparison system. The approach is nonlocal: we give an example where it handles relatively large delays and yields an invariant set containing a large periodic trajectory.

Keywords: Control of linear systems, Control of nonlinear systems, Nonlinear dynamics
Abstract: We investigate the potential use of impulsive inputs for linear and bilinear systems with delays. To set the ideas, we first consider ODE systems. Of interest is the extension to time-variant linear systems, where besides the jump effect on the states we also make concrete investigations of the microscopic singular behavior of the state. We then discuss impulsive bilinear systems. Here it is found that one cannot proceed in a distributional sense. New generalized functions, introduced by Colombeau are discussed. In particular, we settle some issues about the ‘right’ interpretation for extending Schwartz’s theory of distributions and applications to differential equations. Armed with this new perspective we investigate linear time-invariant (LTI) delay systems and classes of bilinear delay systems with impulsive inputs, and define new relevant reachability matrices along the way.